Standard Deviation Calculator
Compute population and sample standard deviation, variance and standard error.
About Standard Deviation Calculator
Standard deviation measures how spread out numbers are around their mean. Use the sample formula (dividing by n − 1) when your data is a sample of a larger group, and the population formula (dividing by n) when you have the entire population.
Formulas
Mean
Mean = (sum of all values) ÷ n
- x̄
- Arithmetic mean
- xᵢ
- Each individual value
- n
- Number of values
Sample variance (Bessel's correction)
Sample variance = Σ(xᵢ − mean)² ÷ (n − 1)
- s
- Sample standard deviation
- x̄
- Sample mean
- n
- Number of values
Population variance
Population variance = Σ(xᵢ − mean)² ÷ n
- σ
- Population standard deviation
- x̄
- Population mean
- n
- Total number of values
Standard deviation
Standard deviation = √variance
Worked example
With the default values below, Standard Deviation Calculator returns Standard Deviation of 13.4907. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Numbers (comma or space separated)
- 4, 8, 15, 16, 23, 42
- Typeone of Sample (n − 1), Population (n)
- Sample (n − 1)
Results
- Standard Deviation
- 13.4907
- Mean
- 18.0000
- Variance
- 182.0000
- Count
- 6
Step by step
- 1Collect the values= 6 values
- 2Calculate the mean= 18.000000
- 3Compute each squared deviation (xᵢ − mean)²
- 4Sum the squared deviations, then divide by n − 1 (5)= 182.000000Sample formula (Bessel's correction): divide by n − 1 to get an unbiased estimate.
- 5Take the square root to get the standard deviation= 13.4907
Assumptions and limits
- The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Standard Deviation Calculator — full guide
Two lists, same mean, different stories
A class where everyone scores 70 and a class where half score 40 and half score 100 both average 70. Standard deviation is the number that separates them: it reports the typical distance between a value and the mean. A production engineer checking whether a filling machine has started to drift, a teacher asking whether a test actually spread students out, and anyone describing the consistency of monthly sales all need that distance, not another average.
Pick sample or population first
Population divides by n and assumes your list is the whole group — every machine on the floor, every student in the room. Sample divides by n − 1 and treats the list as a draw from something larger, so the answer is an estimate of that larger group's spread. Sample is the default because most lists people paste in are samples.
The four steps
- Add the values and divide by the count to get the mean.
- Subtract the mean from each value to get its deviation.
- Square every deviation and add the squares.
- Divide that total by
n − 1(sample) orn(population), then take the square root.
The figure before the square root is the variance. Taking the root returns the answer to the original units, which is why standard deviation is the number you quote.
Worked example with the default list
The defaults are 4, 8, 15, 16, 23, 42. They sum to 108 across 6 values, so the mean is 18.
| Value | Value − 18 | Squared |
|---|---|---|
| 4 | −14 | 196 |
| 8 | −10 | 100 |
| 15 | −3 | 9 |
| 16 | −2 | 4 |
| 23 | 5 | 25 |
| 42 | 24 | 576 |
The squares total 910. Sample variance is 910 ÷ 5 = 182, and the square root of 182 is 13.49. Switch to population and the variance is 910 ÷ 6 = 151.67, giving 12.32 — smaller, as dividing by the larger count always is.
Reading the answer
13.49 against a mean of 18 is a wide spread: the typical gap is about three quarters of the average value itself. For data shaped like a bell curve, roughly two thirds of values sit within one standard deviation of the mean, here 4.51 to 31.49. This list is not bell-shaped — one value, 42, does most of the work — so treat that rule as a sanity check rather than a conclusion.
Where it misleads
- Squaring lets outliers dominate. Drop 42 and the sample standard deviation on the remaining five numbers falls from 13.49 to 7.40.
- It measures spread around the mean, so for skewed data compare it against the median from the average calculator.
- Variance is in squared units — squared rupees, squared minutes — and is awkward to report on its own.
- One list must hold one unit. Mixing minutes with seconds produces a real number and a meaningless result.
To state the spread as a share of the mean, divide one by the other using the percentage calculator.
Frequently Asked Questions
Sample vs population — which do I pick?
If your numbers are a subset drawn from a bigger group, use Sample. If they represent every member of the group, use Population.
Why divide by n − 1 for a sample?
Dividing by n − 1 (Bessel's correction) removes bias when estimating the spread of the full population from a limited sample.
If I add the same amount to every number, does the standard deviation change?
No. Adding 10 to each of 4, 8, 15, 16, 23, 42 moves the mean from 18 to 28, but every deviation from the mean is unchanged, so the sample standard deviation stays 13.49. Multiplying is different: double every value and the standard deviation doubles too, to 26.98.
What happens if I enter just one number in sample mode?
Sample variance divides by n − 1, which is 0 for a single value, so there is no sample standard deviation to report. Switch Type to Population and a one-value list gives 0, since the value equals its own mean. You need at least two numbers for a meaningful sample result.
Should I quote the variance or the standard deviation in a report?
Quote the standard deviation. For the default list it is 13.49 in the same units as the data, while the variance is 182 in squared units. Variance is mainly an intermediate step, though statistical tests such as ANOVA work with it directly, which is why both are shown.
My standard deviation came out larger than my mean — is that a mistake?
It is perfectly possible. The default list gives 13.49 against a mean of 18, about 75% of the mean, because 42 sits far above the rest. It happens routinely with skewed data, with small counts, and with any list containing both positive and negative values.